RU2723970C1 - N-channel linear converter of electromagnetic signals and method of realizing multichannel linear conversion - Google Patents

Abstract

FIELD: physics.SUBSTANCE: invention relates to methods of creating devices which perform linear conversion of electromagnetic signals between a large number of channels. Invention is an N-channel linear converter of electromagnetic signals, where N>2, comprising N channels for signals and M signal mixing units, each of which includes N inputs and N outputs and is characterized by a transfer matrix with complex elements of modulus less than 1, wherein mixing units are connected in series, and at least one input and one output of at least one mixing unit comprises a phase shift element, wherein the mixing units are made with transfer matrices, for which the ratio of at least two corresponding elements of the transfer matrices of at least two mixing units is different from 1, and at the input of at least one mixing unit the number of phase shift elements does not exceed N-2 and the signal conversion method used by the converter.EFFECT: reduced density of arrangement of tunable elements, leading to reduced effect of undesirable cross-interactions.15 cl, 1 tbl, 12 dwg

Description

FIELD OF THE INVENTION

The invention relates to methods for creating devices that perform linear transformations of electromagnetic signals between a large number of channels. The invention can be used in the manufacture of individual elements of communication and computer networks serving a large number of subscribers and computing nodes; these networks can be both classical and quantum. Also, the invention can be used to create devices for the analysis and synthesis of spatial multimode light fields.

State of the art

Linear transformations of electromagnetic signals are used in basic research, and they are also indispensable when creating application devices. They play a large role in the implementation of many applied approaches for processing, receiving and transmitting information. Linear transformations between two signals can be implemented using one divider and two phase shift elements - the basic basic operations for which there are practical implementations in different types of systems.

The prior art method for implementing linear transformations between two channels, the configuration of which can be changed arbitrarily, thus obtaining all possible transformations of a given dimension. This method uses a two-channel interferometer consisting of two series-connected balanced static elements of dividers and two elements of phase shifts, which can be varied to obtain the desired transformation (M. Rack et al., “Experimental implementation of any discrete unitary operator” // Phys. Rev. Lett T. 73, No. 1, S. 58 (1994)).

The disadvantage of this method is the high sensitivity of the conversion quality to deviations of the static elements of the dividers from the balanced values typically occurring when these elements are put into practice. The result of the deviations is the inability to carry out any two-channel conversion, which is associated with the loss of universality of the device.

There is a universal method for implementing multichannel linear transformations by combining two-channel transformations with each other, disclosed in the works of M. Rack et al, “Experimental implementation of any discrete unitary operator” // Phys. Rev. Lett. T. 73, No. 1, S. 58 (1994) and V.R. Clemens et al., “Optimal Design of Universal Multiport Interferometers” // Optica, T. 3, No. 12, P. 1460 (2016). This method allows the construction of arbitrary linear transformations, connecting together the two-channel transforms mentioned above, and the proper selection of the parameters of these two-channel transforms. The essence of the method is to connect one after another many layers of transformations, each of which contains many two-channel transformations acting independently from each other. FIG. 1 illustrates a planar array scheme for cases of an even number of channels N (a) and an odd number of channels N (6), proposed by V.R. Clemens et al., “Optimal Design of Universal Multiport Interferometers” // Optica, T. 3, No. 12, P. 1460 (2016). In the scheme, N layers (1) can be distinguished, each of which is composed of several 2-channel blocks (2) performing simultaneous conversion. The number of 2-channel blocks (2) in each layer depends on the total number of circuit channels N and on the ordinal index of the layer. With even N, the number of 2-channel blocks in each layer is the same and equal to (N-1) / 2. With odd N, the number of 2-channel blocks in a layer varies from layer to layer and can take the values N / 2 and N / 2-1. The total number of blocks making up the multichannel circuit is in all cases N (N-1) / 2. The 2-channel unit used in such multi-channel circuits is shown in FIG. 2 and is a Mach-Zander interferometer consisting of two static balanced dividers and two variable phase shifts - θ and ϕ.

The disadvantage of this universal method is the high sensitivity of the quality of the multi-channel conversion to deviations in the parameters that make up the two-channel conversion. The result is increased requirements for the quality of manufacturing circuits and a decrease in the class of available multi-channel transformations, as a result of which it is impossible to implement arbitrary transformations of a given dimension. In FIG. Figure 3 shows a diagram of a non-ideal 2-channel element of the Mach-Zander interferometer with static dividers characterized by unbalance parameters α and β.

A known method of reducing the influence of errors arising in two-channel transformations on the quality of the conversion proposed in the work of D.A. Miller, “Ideal optics with non-ideal components” // Optica, Vol. 2, No. 8, P. 747 (2015). This method is based on the modification of two-channel interferometers, of which the optical circuits are composed, constructed by known methods. Upon modification, in each of the two-channel interferometers, the static elements of the dividers are replaced by two-channel interferometers. As a result, the interferometer circuit contains additional elements, and the depth of the circuit increases by a factor of 2, and with it the number of variable elements increases by a factor of 2.

The disadvantage of this method is the need to use additional transformation elements, as a result of which the number of elements and the depth of the circuit increases, complicating the practical implementation of the entire transformation. With increasing depth of the circuit, the losses and the area occupied by it increase. In addition, reconfiguration of such circuits requires varying a larger number of elements, which makes monitoring the circuit difficult.

Known multichannel linear Converter and a method for implementing an arbitrary multichannel linear transformation, proposed by J. Zhou, J. Wu, Q. Hu, “Tunable arbitrary unitary transformer based on multiple sections of multicore fibers with phase control)) // Optics Express, vol . 26, No. 3, 3020 (2018). The method is based on the use of multi-channel blocks of constant transformations, which are regions of closely spaced waveguides connected to each other, and layers of variable phase raids located between the blocks, before the first and after the last of them.

A disadvantage of the known solution is the need to simultaneously control a large number of phases in each phase layer, which leads to the need to place variable phase elements in all conversion channels parallel to each other. As a result of this, the geometry of the circuit requires the dense placement of a large number of elements of phase shifts close to each other, leading to an increase in undesirable mutual interaction. Another disadvantage of this method is the difficulty in manufacturing variable phase elements in three-dimensional waveguide circuits, because this involves placing these elements under the outer layer of the waveguides.

A known method of implementing multi-channel linear optical conversion, based on the propagation of multimode optical fields through free space and their conversion using phase light modulators, proposed in Yu. Wang et al. ((Programmable holographic technique for implementing unitary and nonunitary transformations)) // Phys. Rev. A, vol. 95, 033827 (2017).

The disadvantage of this method is the low efficiency of multichannel conversion, expressed, in particular, in a high level of losses. Another disadvantage is its impracticality in implementation.

There is a method of implementing multi-channel linear conversion, which allows you to sort the optical signals received at the input of the conversion into output channels in accordance with the angular momentum of the input signal, which is proposed in the invention of CN 106301595 B. This method uses a cascade of multi-channel transforms, each stage of which can consist of blocks transformations of smaller dimension, acting simultaneously and independently from each other.

The disadvantage of this invention is the ability to implement only a narrow class of multichannel transformations, which makes it incapable of performing arbitrary multichannel transformations.

Closest to the claimed technical solution is a device and method for implementing arbitrary transformations proposed by R. Tang, T. Tanemura, Y. Nakano, Integrated reconfigurable unitary optical mode converter using MMI couplers, IEEE Photonics Technology Letters, vol. 29, No. 12, 971-974 (2017), which is based on the implementation of constant multi-channel conversion units in the form of planar integrated optical circuits and variable phase incursions located in parallel between the constant units, as well as before the first and after the last of them.

The disadvantage of this method is the use of only a certain type of permanent multichannel blocks in the form of planar integrated optical multimode interference elements and variable phase layers in the amount of one less than the total number of conversion channels. As a result of phase constraints, the geometry of the circuit requires a dense arrangement of a large number of elements of phase shifts close to each other, leading to an increase in undesirable mutual interaction.

The technical problem solved by the claimed invention is the high density of the phase shift elements, which leads to a negative effect of mutual influence between closely spaced elements in systems based on spatial channel coding. In such systems, a large number of phase shift elements leads to a high concentration of released power, which imposes high requirements on its resistance to thermal and / or electrical influences. In the case of systems based on time or frequency channel coding, the technical problem to be solved is the need to use a large number of components implementing phase shifts simultaneously, as well as the necessary high frequency of their sequential use, which is problematic with a large number of channels.

Disclosure of invention

The technical result of the invention is to reduce the density of the location of the variable phase elements in circuits with spatial coding and thereby reduce the mutual influence that occurs in the form of thermal, electrical stress or mechanical stress between closely spaced elements when used for phase shifts of thermo-optical, electro Optical or piezoelectric effects, respectively. In addition, a decrease in the density of elements allows to reduce the maximum concentration of thermal and / or electrical energy and / or energy of mechanical stresses in the circuit. This circumstance allows you to increase the maximum possible number of channels of the linear Converter, which can be implemented using the technology used. A consequence of the reduction in energy concentration in the circuit is also an increase in the service life of multichannel linear converters.

In systems with time and frequency coding, the claimed solution allows to reduce the number of simultaneously working components of phase shifts, which makes it possible to reduce their used number. In addition, for such systems, the invention allows to reduce the frequency of operation of the components of the phase shifts, thereby reducing the requirements for them and increasing the maximum possible number of channels of the linear Converter, which can be implemented using the technology. The consequence of reducing the frequency of use of optical elements is also an increase in the service life of devices of multichannel linear converters.

The technical result is achieved through the use of an N-channel linear converter of electromagnetic signals, where N> 2, including N channels for signals and M signal mixing blocks, each of which includes N inputs and N outputs and is characterized by a transfer matrix (or scattering matrix) with complex elements modulo less than 1, while the mixing units are connected in series, and at least one input and one output of at least one mixing unit has a phase shift element, while the mixing units are made with transfer matrices, for which the ratio of at least two corresponding elements of the transfer matrices, at least two mixing units is different from 1, and at the input of at least one mixing unit the number of phase shift elements does not exceed N-2.

The number of channels and mixing units can be selected based on their ratio: N≤M≤N ² -2. The number of channels and mixing units in one of the possible embodiments of an N-channel linear converter is N ² -2, while all phase shifts are placed on one channel.

The mixing unit may be selected with a transfer matrix in which the sum of the squares of the unit modules in each column does not exceed 1 and in each row does not exceed 1. A unitary matrix can be selected as the transfer matrix. The best result is achieved when the columns in at least one transfer matrix are pairwise orthogonal and the rows are pairwise orthogonal. Mixing units can be characterized by transfer matrices with randomly distributed elements.

An N-channel linear converter can be made in the form of a planar or volume integrated optical circuit, where optical waveguides are selected as channels, and thermo-optical or electro-optical or piezoelectric phase-shifting elements are selected as phase shift elements. As channels, frequency or time channels can be used. Channels can be implemented in the form of waveguides or transceivers in free space. Optical fibers, or coaxial cables, or hollow waveguides can be used as waveguides.

The technical result is also achieved in a method for performing an N-channel linear transformation, specified by a linear transformation matrix U ₀ , on an electromagnetic signal, including:

до

;A. Selection of M mixing units characterized by transfer matrices with elements whose modulus lies in the range from

before

;

B. Serial connection of selected blocks in random order;

C. Selection of phase shift elements in an amount not exceeding (N-2) (M + 1);

D. Placement of phase shift elements at the input and / or output of the mixing units on one or more channels (no more than N-2), chosen arbitrarily;

E. Determination of the transfer matrix U _{1 of the} cascade connection of the mixing units with the elements of the phase shift;

F. Determination of phase shift values on each of the phase shift elements by calculating the coordinates of the global maximum of the function

requiring the determination of the transfer matrix U ₁ from step E several times, where Tr (A) denotes the operation of taking a trace of matrix A;

G. Providing a phase shift on the elements of the phase shift of the obtained cascade connection of the mixing units in accordance with the obtained values;

H. The supply of an electromagnetic signal to the input of the obtained cascade connection with the specified values of the phase shift to obtain the converted electromagnetic signal at the output in accordance with the given linear transformation matrix U ₀

The transfer matrix U ₁ , the cascade connection of the mixing units with the elements of the phase shift is determined by the formula:

, где

- фаза, внесенная в j-й канал слоя с индексом m.where V _m is the transfer matrix of the block with index m, Ф ^(m) is the transfer matrix of the phase layer with index m, having the diagonal form

where

is the phase introduced into the jth channel of the layer with index m.

The transfer matrix U _{1 of the} cascade connection of the mixing units with the elements of the phase shift is determined by measuring - applying test signals to the input of the cascade connection and measuring the signals at the output.

The phase shift on the elements of the phase shift is provided by applying a control voltage to the elements of the phase shift, causing a local change in the refractive index in the region of this element due to the thermo-optical, electro-optical or piezoelectric effect.

Thus, the technical result is achieved by reducing the number of phase shift elements located in one phase layer and their arbitrary location in it, which is accompanied by a decrease in the density of the allocated power in a circuit with a spatial channel coding method or the frequency of use of active elements in circuits with temporary and frequency coding. The technical result is also achieved due to the possibility of choosing static mixing units from a large set, characterized by an infinite set of transfer matrices, using the criterion of minimum losses or other characteristics.

Brief Description of the Drawings

In FIG. 1-6 shows explanatory of the prior art images and diagrams, namely:

In FIG. 1, planar schemes of universal multichannel linear conversion for the case of N channels, consisting of N layers (1), each of which consists of independent transformation units (2) that perform conversions between two channels, are shown in the prior art. In FIG. 1a is a diagram with an even number of channels N, and in FIG. 1b - with an odd number of channels N. Positions in the drawings indicate: 1 - layer circuit multichannel linear conversion, 2 - block two-channel conversion from layer 1.

In FIG. 2 schematically shows an embodiment of a transform unit (elementary two-channel) 2 (a). In FIG. 2 (b) a conversion unit is presented, including a divider 4, characterized by an angular parameter θ that determines the division coefficient, and a phase shift element 3 ϕ in one of the channels. In FIG. 2 (c) presents a prior art embodiment of a reconfigurable conversion using an interferometer (5), using two constant elements of a balanced divider 4 and two variable phase shift elements 3, with which you can control the transformation parameters θ and ϕ.

In FIG. 3 shows a prior art embodiment of a conversion unit in the form of a reconfigurable Mach-Zander interferometer (5) with constant elements of dividers 4, which is an element (2) of multi-channel circuits created by known methods. In a real situation, the elements of the dividers are not balanced, as known methods require. The degree of imbalance of these elements is indicated by the parameters α and β.

In FIG. 4 shows histograms of the distribution of the degree of correspondence of the conversion of an 8-channel interferometer constructed according to the well-known scheme depicted in FIG. 1, composed of two-channel blocks 6 with imbalances introduced. Histograms were obtained by generating 1000 arbitrary unitary matrices and numerically searching for the parameters of the variable phases for each of the 2-channel blocks that correspond to the maximum measure of compliance. 3 imbalance models were used (see text below).

In FIG. 5 depicts a well-known multi-channel optical scheme using, as constant multi-channel mixing blocks, the communication regions between closely spaced waveguides (6) and phase layers (7) containing N-1 phases, where N is the number of conversion channels.

In FIG. Figure 6 shows the architecture of multi-channel circuits using constant mixing units V _j (8), in the general case, performing transformations between all N channels, and layers of phase shifts Ф ^(m) (7), which can vary N-1 phases. Permanent multi-channel blocks and variable phase layers are arranged alternately.

In FIG. 7-12 are drawings illustrating the claimed invention:

In FIG. Figure 7 shows the architecture of multi-channel circuits using constant mixing units V _j (8), in the general case, performing transformations between all N channels, and layers of phase shifts Ф ^(m) (9), which can vary N-2 phases. Permanent multi-channel blocks and variable phase layers are arranged alternately.

In FIG. 8 shows a multi-channel circuit with constant mixing units 8, in which the number of variable phases in the phase layer (10) and their placement in each layer can be arbitrary. The number of multi-channel blocks 8 in this circuit is N + K, and the number of phase layers is N + K + 1, where K is determined by the number of variable phase parameters necessary for the universality of the circuit.

In FIG. 9 shows a multi-channel circuit with constant blocks 8, in which the number of variable phases in each phase layer (11) is 1, and the number of such layers is N ² - 1.

In FIG. 10 shows histograms of the distribution of the conversion quality parameter of the interferometers depicted in FIG. 5, composed of permanent multi-channel blocks taken at random. Histograms were obtained by numerical optimization of variable phase layers for a set of 1000 random unitary matrices, for each of which constant blocks were also generated randomly. The number of transformation channels was chosen N = 10 (a) and N = 30 (b).

In FIG. 11 shows histograms of the conversion quality parameter of the multi-channel interferometer shown in FIG. 9 for N = 10 and N = 20 with the number of phase layers N ² - 1.

In FIG. 12 a, b and c show three types of tunable three-channel interferometer circuits corresponding to the general circuits shown in FIG. 7, FIG. 8 and FIG. 9, respectively, which were implemented by the authors by the method of femtosecond laser printing.

The implementation of the invention

The invention relates to transducers of electromagnetic fields, which can belong to different wavelength ranges - from radio to optical.

For a more unambiguous understanding of the essence of the claimed invention, the following are the main terms and definitions used in the framework of the present description.

A conversion channel is called any degree of freedom of the electromagnetic field, which can be uniquely associated with an independent signal characterized by a complex amplitude. For clarity, we separately distinguish the following degrees of freedom, which can be a transformation channel: 1) the spatial degree of freedom of the field can be a mode of a single-mode waveguide, along which an electromagnetic signal can propagate. In this case, several single-mode waveguides form many channels. Also, the spatial channel may be the spatial one mode of a multimode waveguide or one mode of free space. In this case, several spatial modes of one multimode waveguide or several modes of free space form many independent channels. 2) The frequency degrees of freedom of the field are individual lines of the frequency spectrum. A set of non-overlapping spectral lines forms many channels. 3) A temporary degree of freedom is a given period of time, characterized by a temporary report of its beginning and end. The presence of an electromagnetic signal in this period of time is interpreted as the presence of a signal in this channel. Thus, a set of time segments forms many channels. In accordance with which of the indicated degrees of freedom of the electromagnetic field is used to encode information, they talk about spatial, frequency and time channels.

A linear N-channel transform is a transform between N channels, the action of which can be described by a linear law:

- комплексные амплитуды на входе преобразования,

- комплексные амплитуды на выходе преобразования. Здесь индексы j и k принимают значения от 1 до N. В(1.1) комплексные коэффициенты U_kj формируют матрицу U размерности N×N, которая и определяет конкретное линейное преобразование. Выражение (1.1) может быть представлено в матричном виде:where N is the number of conversion channels,

- complex amplitudes at the input of the conversion,

- complex amplitudes at the output of the conversion. Here, the indices j and k take values from 1 to N. In (1.1), the complex coefficients U _kj form an N × N matrix U, which defines a specific linear transformation. Expression (1.1) can be represented in matrix form:

и

- столбцы, составленные из амплитуд сигналов на входе и выходе преобразования, соответственно. Число каналов преобразования N характеризует размерность преобразования. Изобретение относится к случаю, когда число каналов преобразования N≥3.Where

and

- columns composed of the amplitudes of the signals at the input and output of the conversion, respectively. The number of transform channels N characterizes the dimension of the transform. The invention relates to the case when the number of conversion channels is N≥3.

The transfer matrix of the transformation, or simply the transformation matrix, is called the matrix U, which relates to each other a column of amplitudes at the output of the transformation with a column at its input (see expression (1.2)).

An N-channel linear converter or a linear N-channel device or a linear N-channel interferometer is any device that performs linear N-channel conversion of electromagnetic signals.

A multichannel conversion mixing unit is an N-channel static element in an N-channel interferometer whose matrix has at least N-1 non-zero elements in each row and column. Two multichannel mixing units connected in series with each other, between which there is no variable phase shift element, are considered as one mixing unit in the framework of the present invention.

An N-channel transform as part of an N-channel interferometer whose matrix has a diagonal form is called a phase conversion layer or simply a phase layer.

those. it performs an independent phase shift by ϕ _j in the channel with index j. However, not all N channels of the phase layer may contain phase shifts. In this case, there are units in the diagonal elements in (1.3) corresponding to channels without phases.

. В матричном виде они принимают вид:If there is no loss in the transformation, then its matrix is unitary, i.e. for it the relations

. In matrix form, they take the form:

, отражающее сохранение энергии при осуществлении преобразования. В общем случае в преобразовании имеются потери и соотношение (1.4) не выполняется. Вместо этого справедливо используют более общее представление

.where "*" means the operation of complex conjugation of the matrix, I is the identity matrix. Condition (1.4) implies the equality:

, reflecting the conservation of energy during the implementation of the conversion. In the general case, there are losses in the transformation and relation (1.4) is not satisfied. Instead, fairly use a more general view

.

To compare the two transformations that are described by the matrices U ₁ and U ₂ , use the following measure of correspondence:

where Tr (A) denotes the operation of taking a trace of matrix A. For lossless transformation (its matrix is unitary), Tr (U * U) = N. The maximum value of F ^(loss) = 1 corresponds to the ideal match of two matrices. The minimum value of F ^(loss) = 0. In the future, cases of the implementation of unitary transfer matrices in interferometer circuits will be considered. More general cases of non-unitary matrices (linear transformations with losses) can be considered as schemes that are part of larger unitary schemes.

Using correspondence measure (1.5), the ideal value F ^(loss) = 1 can be obtained for two matrices, which, generally speaking, are not equal to each other U ₁ ≠ U ₂ , but between them the relation U ₂ = tU ₁ holds, where t - arbitrary complex number. The module | t | it makes sense for the overall transmittance (in amplitudes) of the conversion and for | t | <1 it corresponds to a conversion with balanced losses - when the signal is applied to any input, it experiences the same losses in any of the outputs, i.e. This situation corresponds to balanced losses. The presence of a phase at t leads to the insensitivity of measure (1.5) to the general phase of the transformation. Thus (1.5) is independent of losses if they are balanced and the general phase of the compared transformation is also not important. Thus, the losses in the circuit are considered balanced (its matrix is U = tU ₀ , where U ₀ is unitary, a | t | <1).

The invention proposes a different universal multi-channel linear conversion scheme, which differs from the known schemes shown in FIG. 5 and FIG. 6, in that the number of variable phase elements in the phase layer can be any, and their placement along the channels can be arbitrary. In addition, multichannel mixing units can be selected from a wide class characterized by a large number of possible transfer matrices. The latter circumstance allows us to choose as mixing blocks, based on considerations of the minimum loss in these elements or the optimum of another important characteristic.

Schematic representations of possible circuit configurations proposed in the present invention are shown in FIG. 7, FIG. 8 and FIG. 9. In contrast to the circuits shown in FIG. 5 and FIG. 6, the number of phases in each of the phase layers can be less than N-1; therefore, the number of constant multi-channel blocks (mixing blocks) is greater than N.

In FIG. 7 depicts a variant of a multi-channel conversion scheme using generally different mixing blocks V _m and N-2 phases in each phase layer (9). In FIG. Figure 8 shows a variant of a multi-channel conversion scheme in which the number of phases in each phase layer (10) may be different, as well as their placement in this layer. In FIG. Figure 9 shows a multi-channel conversion scheme using only one phase in each phase layer (11), thus, the number of mixing blocks in this case is the maximum possible for a given number of channels N and is equal to N ² - 2. In all these variants of the proposed architecture of universal multi-channel schemes interferometers, the total number of circuit parameters N ² - 1 necessary for the implementation of an arbitrary unitary conversion is achieved by increasing the number of constant multi-channel blocks and phase layers.

The transfer matrix of the N-channel transform in this case takes the following form:

where V _m is the transfer matrix of the mixing block with the index m, Ф ^(m) is the transformation matrix of the phase layer with the index m, which includes phase elements in an amount less than N-1. In FIG. 8 and in formula (1.6), the integer K characterizes how much the resulting circuit is deeper (i.e. longer / consists of a larger number of layers) than the circuits shown in FIG. 5 and FIG. 6.

Note that the conversion matrix of the N-channel interferometer constructed according to the known methods depicted in FIG. 5 and FIG. 6 is as follows:

where the diagonal matrix of each phase layer Ф ^(m) contains exactly N-1 phase shifts, and the number of mixing units in the circuit is exactly N.

The choice of transformations of the constant multichannel blocks V _m determines the possibility of implementing arbitrary matrices. Indeed, not all V _m provide universality to the multichannel scheme (1.6). For example, if all V _m have the form of a unit matrix, then there will be no interaction between the channels in (1.6) and the whole circuit will perform phase transformation with the diagonal matrix diag (exp (iϕ ₁ ), exp (iϕ ₂ ), ..., exp ( iϕ _N )).

Below is the rationale for the use of a wider class of linear transformations when choosing multichannel mixing units than was used in other methods for creating universal multichannel interferometers.

, которые при заданных передаточных матрицах блоков смешения V_m (

, где М=N+K) и размещении этих фаз в фазовых слоях наилучшим образом реализуют заданную матрицу U₀, использовался численный алгоритм, который находил максимум функции (1.5) с матрицей (1.6) и U₂=U₀ в пространстве всех фаз

. При заданной размерности многоканальной схемы N генерировались унитарные матрицы U₀ случайным образом согласно мере Хаара (М. Лундберг и Л. Свенсен «Мера Хаара и генерация случайных унитарных матриц», IEEE Sensor Array and Multichannel Signal Processing Workshop, 114, 2004) в количестве 1000 штук. Для каждой матрицы U₀, которую необходимо было реализовать в многоканальной схеме с помощью подбора надлежащих фаз

, генерировались N матрицы V_m постоянных блоков также случайно согласно мере Хаара. Далее наборы фаз

находились с помощью численного алгоритма поиска глобального максимума функции (1.5).To find the values of tunable phases

which, for given transfer matrices of the mixing units V _m (

, where M = N + K) and the placement of these phases in the phase layers best implement the given matrix U ₀ , a numerical algorithm was used that found the maximum of function (1.5) with matrix (1.6) and U ₂ = U ₀ in the space of all phases

. For a given dimension of a multichannel scheme N, unitary matrices U _{0 were} randomly generated according to the Haar measure (M. Lundberg and L. Swensen, “Haar Measure and Generation of Random Unitary Matrices”, IEEE Sensor Array and Multichannel Signal Processing Workshop, 114, 2004) in the amount of 1000 pieces. For each matrix U ₀ that needed to be implemented in a multichannel scheme by selecting the appropriate phases

, N matrices V _{m of} constant blocks were generated also randomly according to the Haar measure. Further sets of phases

were found using the numerical algorithm for finding the global maximum of function (1.5).

In FIG. 10 shows the correspondence measures obtained in the numerical analysis of the histogram (1.6), which illustrate the conversion quality of a multichannel interferometer, the circuit of which is shown in FIG. 6 for the case of random transfer matrices V _m . The obtained histograms are drawn for circuits with the number of channels N = 10 (a) and N = 30 (b). The distributions shown in one graph differ in the number of phase layers (and blocks of constant transformations) that make up the circuit. For the case N = 10, the number of phase layers is 2, 5, 8, and 11. The corresponding number of circuit parameters is 18, 45, 56, and 99. For the case N = 30, the number of phase layers is 7, 15, 26, and 31, which corresponds to the number of parameters 203, 435, 754 and 899, respectively. As can be seen from the graphs, if the number of layers in the circuit is less than N + 1, then the conversion quality parameter does not reach the ideal value F = 1. However, if the number of layers reaches N + 1, which is necessary for the total number of parameters N ² -1, the quality of the circuit conversion is ensured with high accuracy F = 1 (all obtained values in numerical calculations F> 0.99999).

A similar numerical study was carried out for all variants of the optical schemes of interferometers depicted in FIG. 7, FIG. 8 and FIG. 9. In FIG. 11 is a graph showing histograms of the correspondence parameter (1.5) for an interferometer with one phase in the phase layer (Fig. 9) for dimensions N = 10 and N = 20. As can be seen from the graph, for a larger proportion of matrices, the correspondence parameter F> 0.999. The dependences of the correspondence parameter (1.5) for the other two variants of the schemes are of a similar nature. It should be noted that the results obtained with F <1 are related to the finite accuracy of the implementation of the numerical optimization algorithm.

и в общем случае осуществляющих преобразование между N каналами. В рассматриваемых многоканальных схемах постоянный блок преобразования может состоять из нескольких соединенных последовательно постоянных блоков, области соединения которых не содержат варьируемых фаз.Arbitrary unitary linear transformations using constant multi-channel transform blocks can be implemented in accordance with the circuit shown in FIG. 5. The N-channel circuit consists of constant multi-channel blocks 6 described by transfer matrices V _m

and in the General case, carrying out the conversion between N channels. In the multichannel circuits under consideration, the constant conversion unit may consist of several constant units connected in series, the connection areas of which do not contain variable phases.

. В таких схемах каждый из фазовых слоев содержит в точности N-1 варьируемых фаз.The claimed solution allows the implementation of arbitrary multichannel linear optical transformations presented in the diagram of FIG. 5, which differ in the specific form of transfer matrices of constant blocks V _m

. In such schemes, each of the phase layers contains exactly N-1 variable phases.

A special case of mixing units 8 is transfer matrices 6, which can be implemented in the form of creating areas of proximity of the cores close to each other, so that a connection arises between them (J. Zhou, J. Wu, Q. Hu, “Tunable arbitrary unitary transformer based on multiple sections of multicore fibers with phase control)) // Optics Express, vol. 26, No. 3, 3020 (2018)).

The interaction of the field amplitudes in the region of closely spaced waveguides, which has a certain length L, can be described by a system of equations:

- вектор амплитуд полей в точке с координатой z, амплитуды

отвечают значениям в начале области взаимодействия, М - матрица, описывающая связь между волноводами, конкретный вид которой зависит от размещения волноводов друг относительно друга. Например, при круговом размещении со связью между ближайшими соседями матрица М выглядит следующим образом:Where

- vector of field amplitudes at a point with z coordinate, amplitudes

correspond to the values at the beginning of the interaction region, M is a matrix describing the relationship between the waveguides, the specific form of which depends on the placement of the waveguides relative to each other. For example, in a circular arrangement with a connection between the nearest neighbors, the matrix M looks as follows:

и при заданной конфигурации взаимного расположения волноводов определяется длиной области связи L_k (k=1,2,…,N). Изменение конфигурации матрицы преобразования всего интерферометра осуществляют с помощью слоев преобразований

(m=1,2,…,N+1), располагаемых между постоянными блоками

.where C _mn are the coupling coefficients between the waveguide channels with indices m and n. Thus, the transformation of the kth multichannel block can be written as

and for a given configuration of the mutual arrangement of the waveguides is determined by the length of the communication region L _k (k = 1,2, ..., N). Changing the configuration of the transformation matrix of the entire interferometer is carried out using transformation layers

(m = 1,2, ..., N + 1) located between the constant blocks

.

The claimed solution allows to eliminate the disadvantages of the known methods associated with the need to control a large set of phases in one layer, the minimum number of which is equal to N-1, where N is the number of conversion channels that is difficult to implement in practice .. In particular, when using coding in spatial channels ( waveguide or open-space modes) in three-dimensional arrangement of waveguide channels (see J. Zhou, J. Wu, Q. Hu, “Tunable arbitrary unitary transformer based on multiple sections of multicore fibers with phase control” // Optics Express, vol. 26, No. 3, 3020, 2018), it is necessary to control the phase shifts of each channel separately, for which it is necessary to place the corresponding element (thermo-optical, electro-optical or piezo-optical) in the depth of the circuit behind the outer layer of the waveguides. This drawback in the claimed solution in one embodiment is overcome by bringing the channel placement geometry to planar geometry in the regions of the phase layers, thus providing access to each channel, or spacing in the space of the waveguides. Moreover, this implementation option is associated with an increase in the size of the circuit d for large values of N.

Another advantage of the claimed invention is to overcome the disadvantage of existing analogs using circuits with N-1 elements of phase shifts in one layer, associated with the negative influence of cross-interaction between them, in which closely spaced elements of phase shifts are not completely isolated from each other, which leads to introducing shifts into the channel from neighboring elements. This drawback is overcome in the claimed solution by spacing phase shifts in the space of elements.

Arbitrary multichannel linear optical transformations can be implemented in planar integrated optical structures (see R. Tang, T. Tanemura, Y. Nakano, Integrated reconfigurable unitary optical mode converter using MMI couplers, IEEE Photonics Technology Letters, vol. 29, No 12, 971-974, 2017), with mixing blocks and variable phase incursions located in parallel between the constant blocks, as well as before the first and after the last of them. In integrated optical design, multimode interference elements can be used as blocks of constant multichannel transformations. The principle of operation of such blocks is based on the interference between several signals, which occurs in the waveguide structure.

The claimed technical solution when using the scheme with planar multimode interference blocks also allows you to eliminate the above disadvantages inherent in the optical system based on fibers with many cores, using N-1 elements of phase shifts in one layer. The claimed solution allows you to arrange the elements of phase shifts with the exception of the negative the effects of cross-interaction between them.

Thus, the proposed method for creating a universal multi-channel transform can be implemented in systems with spatial coding. In such systems, a waveguide or a spatial mode of free space can act as a channel. In optics, waveguide structures forming spatial-coded circuits can be made in the form of an integrated optical circuit that can be manufactured using planar lithography technology (L. Chrostowski, M. Hochberg, Silicon Photonics design: from devices to systems // Cambridge Univ. Press, 2015 ) or by three-dimensional technology, for example, laser printing (IV Dyakonov et al., Reconfigurable photonics on a glass chip // Phys. Rev. Applied, vol. 10, 044048 (2018)). As phase shift elements in such schemes, 1) thermo-optical elements can be used that change the phase shift of the waveguide section when electric current is passed through them due to its heating, 2) electro-optical elements that change the phase shift when voltage is applied due to changes in the concentration of electrons and / or holes in the waveguide section, and 3) a piezoelectric element that changes the phase incursion of the signal propagating in the waveguide under the action of an electric voltage that causes mechanical stress in the waveguide section, which changes its refractive index.

The claimed invention can also be implemented in physical systems with frequency coding of channels. In this case, non-overlapping frequency lines of the spectrum of the electromagnetic field act as discrete channels. Each of the lines is assigned a channel number, thus, a set of N lines forms a plurality of channels of the entire conversion. The amplitude of the signal propagating through the frequency channel is the complex amplitude of the corresponding frequency component. It is worth noting that the use of frequency coding of multichannel conversion channels allows the use of one spatial channel for transmission and conversion - one waveguide, one spatial mode of free space. To implement the interaction between frequency channels, it is proposed to use a set of frequency channel modulation transformations, which induces a time-dependent phase on the channels, the profile of which can be selected, and a pulse shaper.

Multichannel systems with frequency channels are most widely used in optics, in particular in quantum. The claimed invention can be implemented by analogy with the system described in the work of N.-N. Lu et al., Electro-optical frequency beam splitters and fritters for high-fidelity photonic quantum information processing // Phys. Rev. Lett., Vol. 120, 030502, 2018), which demonstrated in practice the implementation of Hadamard transforms for one frequency channel and the tritter transform for three frequency channels. To implement the invention can be used fiber optic equipment, which is standard for the telecommunication wavelength range in the region of 1550 nm. To implement modulation, electro-optical modulators and a pulse shaper can be used, which are standard components of telecommunication equipment. In addition to fiber optic telecommunication components, their entire set can also be implemented using integrated optical circuits, as All necessary items have been demonstrated. For example, integrated optical pulse shaping is demonstrated by K.A. McKinzie et al. InP integrated pulse shaper with 48 channel, 50 GHz spacing amplitude and phase control, 2017 IEEE Photonics Conference (IPC), 197-198 (2017). Integrated optical modulators have long been available for manufacture in integrated optical design (K. Ogawa, Integrated silicon-based optical modulators: 100Gb / s and beyond, SPIE Press, ISBN: 9781510625815, 2019).

The claimed invention can also be implemented in systems using time coding. In this case, non-overlapping time intervals act as discrete channels. Each time interval is assigned a channel number. The pulse of an electromagnetic signal located in a time interval with a certain index m is interpreted as a signal in channel m, and the amplitude of a pulse is interpreted as the amplitude of a signal in a channel with this index. As with frequency coding, time coding uses a single spatial channel, which may be a waveguide or spatial mode of free space. To implement the interaction between the channels in this scheme, delay lines and dynamically variable dividers are needed.

In particular, the claimed invention can be implemented by analogy with the multi-channel time-coding scheme presented in K.R. Motes et al., Scalable boson sampling with time-bin encoding using a loop-based architecture // Phys. Rev. A, vol. 113, 120501 (2014), in which a universal multi-channel circuit is proposed, which is based on fiber delay loops and reconfigurable two-channel dividers - these are standard components that are used, for example, in telecommunications. Multichannel conversion units can be implemented using several nested delay loops and constant dividers, while variable phase shifts can be implemented using dynamically programmable phase modulators or phase shifts.

Case Studies

To implement the proposed invention, the authors used the technology of femtosecond laser printing, which allows you to create both planar and three-dimensional integrated optical circuits. Using this technology, optical schemes with the number of channels N = 3 were implemented. To create an integrated chip, a quartz blank in the shape of a rectangle with a length of 10 cm, a width of 5 cm and a thickness of 5 mm was used. At the first stage, waveguide structures were created in the bulk of the preform, forming a static optical scheme containing mixing blocks and sections of waveguides that would realize phase elements. Mixing units were implemented in the form of three alternately connected elements of directional dividers (see Fig. 12). The division coefficient by the power of each divider was determined by the choice of the length of the interaction region and was set randomly at the manufacturing stage and lay in the range from 1/3 to 2/3.

To change the configuration of optical circuits, a traditional method was used, based on the thermo-optical effect - changing the refractive index of a substance under the influence of temperature changes. For this purpose, metal resistive heaters were applied to the surface of the quartz chip above the sections of waveguides in which it is necessary to change the phase incursion of the signals. Under the influence of the potential difference, the resistive heaters were heated. Details of manufacturing technology can be found in I.V. Dyakonov et al., Reconfigurable photonics on a glass chip // Phys. Rev. Applied, vol. 10, 044048 (2018).

Three types of circuits were created and experimentally studied. In FIG. 12 is a schematic representation of fabricated integrated optical circuits: FIG. 12a shows a circuit with 2 phase shift elements in each phase layer and 3 mixing units; in FIG. 12b shows a diagram with a different number of phases in the phase layer (1 or 2 phases) and 5 mixing units; in FIG. 12c shows a circuit with 1 phase in each phase layer and 8 mixing units. Each type of circuit was made in the amount of 5 pieces and the same series of experiments was carried out on all of them. In this case, the phase elements shown not in the output channels in FIG. 12 were not manufactured, because ultimately measured the power of the output signals.

To verify the universality of the fabricated integrated optical circuits and their other characteristics related to the claimed result, the following sequence of experiments was carried out. A unitary matrix U _{0 of} size 3 × 3 was randomly generated, which was necessary to implement in the fabricated schemes by adjusting the variable elements of the phase shifts. For this purpose, at given values of phase shifts (determined by the voltages at the corresponding heating elements), an optical signal of the same power was individually supplied to each of the inputs of the interferometer and measurements were made of the signal power at the output of each channel. Thus, the modules of the transfer matrix elements were determined for the given values of the phase shift. To determine the arguments of complex elements, two signals of the same power with a variable phase delay between them were input. The procedure with paired input signals was repeated for the entire set of 3 pairs of input channels. These actions allowed to restore the transfer matrix U ₁ , tunable circuit. Using information about the matrices, the parameter (1.5) was calculated, the value of which served as the value of the objective function in the optimization algorithm. The optimization algorithm corrected the values of phase shifts. The described procedure was repeated iteratively ~ 1000 times until it completely converged to the global maximum, which is characterized by the value of parameter (1.5) equal to unity. Details of the algorithm used to reconstruct the transfer matrices of multichannel circuits can be found in an article by M. Tillmann, C. Schmidt, P. Walther, “On unitary reconstruction of linear optical networks”, Journal of Optics, vol. 18, 114002 (2016).

The table shows the average values (for 5 implementations of each type of circuit and 100 random matrices U ₀ ) of parameter (1.5) achieved in the experiment, as well as the average values of electric power per phase shift element.

As can be seen from the table, in all cases high values of the parameter F were obtained, which indicates the universality of the proposed schemes. Also note that with a decrease in the density of the arrangement of the elements of the phase shifts (from a to b), the average power per phase shift decreases, which may be a consequence of a decrease in the mutual influence. In addition, it is obvious that the dissipated thermal power per unit area of the circuit decreases.

Claims (27)

1. N-channel linear converter of electromagnetic signals, where N> 2, including N channels for signals and M signal mixing blocks, each of which includes N inputs and N outputs and is characterized by a transfer matrix (or scattering matrix) with complex elements, modulo less than 1, while the mixing blocks are connected in series, and at least one input and one output of the at least one mixing block has a phase shift element, characterized in that the mixing blocks are made with transfer matrices for which the ratio of at least two corresponding elements of the transfer matrices of at least two mixing units is different from 1, and at the input of at least one mixing unit the number of phase shift elements does not exceed N-2. 2. The N-channel linear converter according to claim 1, characterized in that the number of channels and mixing units is selected based on the ratio: N≤M≤N ² -2. 3. The N-channel linear converter according to claim 1, characterized in that the number of channels and mixing units is N ² -2, while all phase shifts are placed on one channel. 4. The N-channel linear converter according to claim 1, characterized in that the mixing unit is selected with a transfer matrix in which the sum of the squares of the element modules in each column does not exceed 1 and in each row does not exceed 1. 5. The N-channel linear converter according to claim 1, characterized in that the unitary matrix is selected as the transfer matrix. 6. The N-channel linear converter according to claim 4, characterized in that, in at least one transfer matrix, the columns are pairwise orthogonal and the rows are pairwise orthogonal. 7. The N-channel linear converter according to claim 1, characterized in that the mixing units are characterized by transfer matrices with randomly distributed elements. 8. The N-channel linear converter according to claim 1, characterized in that it is made in the form of a planar or volume integrated optical circuit, where optical waveguides are selected as channels, and thermo-optical or electro-optical or piezoelectric phase-shifting elements are selected as phase shift elements. 9. The N-channel linear converter according to claim 1, characterized in that frequency or time channels are selected as channels. 10. The N-channel linear converter according to claim 9, characterized in that the channels are implemented as waveguides or transceivers in free space. 11. The N-channel linear converter according to claim 10, characterized in that the optical fibers or coaxial cables or hollow waveguides are used as waveguides. 12. A method of performing an N-channel linear transformation specified by a linear transformation matrix U ₀ over an electromagnetic signal, comprising: a. selection of M mixing units characterized by transfer matrices with elements whose modulus lies in the range from

before

; b. serial connection of selected blocks in random order; c. the choice of phase shift elements in an amount not exceeding (N-2) (M + 1); d. placing phase shift elements at the input and / or output of the mixing units on one or more channels (no more than N-2), chosen arbitrarily; e. determination of the elements of the transfer matrix U ₁ , cascade connection of the mixing units with the elements of the phase shift; f. determination of phase shift values on each of the phase shift elements by calculating the coordinates of the global maximum of the function

requiring the determination of the transfer matrix U ₁ from step E several times, where Tr (A) denotes the operation of taking a trace of matrix A; g. providing a phase shift on the phase shift elements of the obtained cascade connection of the mixing units in accordance with the obtained values; h. applying an electromagnetic signal to the input of the obtained cascade connection with predetermined phase shift values to obtain the converted electromagnetic signal at the output in accordance with a given linear transformation matrix U0. 13. The method according to p. 12, characterized in that the transfer matrix U _{1 of the} cascade connection of the mixing units with the elements of the phase shift is determined by the formula:

where V _m is the transfer matrix of the block with index m, Ф ^(m) is the transfer matrix of the phase layer with index m, having the diagonal form

where

is the phase introduced into the jth channel of the layer with index m. 14. The method according to p. 12, characterized in that the transfer matrix U1 of the cascade connection of the mixing units with the elements of the phase shift is determined by measuring - applying test signals to the input of the cascade connection and measuring the signals at the output. 15. The method according to p. 12, characterized in that the phase shift on the phase shift elements is provided by applying a control voltage to the phase shift elements, causing a local change in the refractive index in the region of this element due to the thermo-optical, electro-optical or piezoelectric effect.

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